The first case leads to self-dual connection 1-forms and Euler-top system, while the second case can be resumed to \(\lambda _1=\lambda _2=\lambda _3=\pm 2\)  by an appropriate change of sign in \(c_i\)  [1]}. Therefore, from equations (REF ) and (REF ) we get
\(\left( \frac{\partial _r c_i}{c_0} \right) = \mp \left( \frac{ c_j^2 + c_k^2 - c_i^2 }{ c_j c_k} -2\right), \qquad \partial _r \big ( \ln c_i^2 \big ) = \mp 2\big ( c_j^2 + c_k^2 - c_i^2 - 2c_j c_k \big ) .\) 
